A signal from a source (like a bass, a mic, a preamp) into an input needs two conductors so the electrons can flow there and back.

In an unbalanced connection, like through an instrument cable with 1/4" connectors, or with "hi-fi" interconnects with RCA (phono) connectors, those conductors are the center wire of the cable and the shield around it. The shield is ground. This works adequately in many situations, but its only protection against noise is the shield, and sometimes that's not enough. The shield can protect the inner conductor against electrostatic noise, but unless it's made of steel or other magnetically permeable metal (very, very rare), it won't protect against electromagnetic noise.

In a balanced connection, there are two signal conductors, neither of which are grounded. The input is differential, meaning that it has two terminals and only responds to differences in the signal voltage between them (differential mode), ignoring any voltages that are the same (common mode). From output through cable to input, it is important that both legs of the balanced connection have equal impedances to ground. These two criteria--differential input and equal impedances to ground--are what makes the connection balanced. Balanced connections are highly immune to noise, and that's why they are used for mics, long signal runs, pro audio interconnects, etc.

The two signal conductors should be twisted together. This serves two purposes: it reduces the loop area between the two conductors, reducing the differential coupling of electromagnetic noise into them; and it gives both conductors somewhat equal exposure to the ambient electrical fields, reducing the differential coupling of electrostatic noise. The equal impedances to ground ensure that any noise currents induced electromagnetically into both conductors will result in equal noise voltages, which will cancel out in the differential input.

In pro and music systems, these balanced cables are also shielded. However, the biggest users of audio circuitry in the world--telephone companies--use millions of miles of unshielded twisted-pair wire with transformer coupling (which ensures very precisely equal impedances and extremely high rejection of common-mode voltages), and their lines are highly immune to noise considering the distances they cover.

The last element of the balanced connection is the signal source. Usually, the source puts signal voltages that are equal but opposite in polarity on the two legs, but their being equal is not actually necessary for a good balanced connection; equal impedances to ground are.